Articles of fourier transform

Discrete fourier transform on $\mathbb{Z}/N\mathbb{Z}$ vanishing on an interval of size at least $\sqrt{N}$

Let $f : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let $\hat{f} : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ be its DFT given by $\hat{f}(m) = \sum_{j \in \mathbb{Z}/N\mathbb{Z}} f(j)e_N(jm)$, where $e_N(x) := e^{2\pi i x/N}$. If $\hat{f}(m) = 0$ for every $m$ in some interval $[M+1, M+K]$ with $K \geq \sqrt{N}$, is there anything interesting we can say about $f$ […]

Principal value not appearing in Fourier transform of the Heaviside function

I am trying to determine the Fourier transform of the Heaviside function $H$ as a tempered distribution. Consequently, for $\phi\in\mathcal{S}(\mathbb{R})$, I get $$\begin{aligned} (\mathcal{F}H)(\phi)&=H(\mathcal{F}\phi) \\ &=\int_0^\infty(\mathcal{F}\phi)(\xi)\,d\xi \\ &=\int_0^\infty\left(\int_{-\infty}^\infty\phi(x)e^{-i\xi x}\,dx\right)\,d\xi \\ &=\int_{-\infty}^{\infty}\left(\int_0^\infty e^{-i\xi x}\,d\xi\right)\phi(x)\,dx\qquad\text{by Fubini} \\ &=\lim_{N\to\infty}\int_{-N}^N\left(\frac{i}{x} e^{-i\xi x}\bigg|_{\xi=0}^{\infty}\right)\phi(x)\,dx \\ &=\lim_{N\to\infty}\int_{-N}^N\left(\frac{i}{x}\lim_{R\to\infty}e^{-i\xi x}\bigg|_{\xi=0}^R\right)\phi(x)\,dx \\ &=i\lim_{N\to\infty}\int_{-N}^N\frac{\phi(x)}{x}\lim_{R\to\infty}e^{-iRx}\,dx-i\lim_{N\to\infty}\int_{-N}^N\frac{\phi(x)}{x}\,dx, \end{aligned}$$ so where am I going wrong? If I had $\epsilon\downarrow 0$ […]

An integral related to the Fourier transform of $\sin(bx)$

Let $a,b >0$ and consider the following integral: $$ I(a,b) = \int_{0}^{\infty} e^{ – i a x } \sin( b x )\ dx $$ Is there a way to evaluate this integral? This looks very similar to the Fourier transform of the function $\sin(bx)$, but the limits of integration are different. Looking at the wikipedia […]

What is the Fourier Transform for $\mathscr{F} \left\{\frac{\partial^2 (x^2p(x,t))}{\partial x^2} \right\}$ w.r.t $x$?

What is the Fourier Transform for the following: $$ \mathscr{F} \left\{\frac{\partial^2 (x^2p(x,t))}{\partial x^2} \right\} = ? $$ Here is my problem: Suppose we are given the diffusion $$ dX(t)= \mu dt + \sigma dW(t) \hspace{10mm} (1) \\ X(0)=x_0 $$ And a function $$ k(x)= \theta x^2 \hspace{10mm} (2) $$ Then by Ito Lemma $$ dk[X(t)]=(2 […]

Identity for all uniformly bdd functions follows “by basic Fourier (transform) analysis”?

I found an interesting identity in a paper in a paper I am reading (page $9$). The statement is as follows: $\mathcal F$ is the Fourier transform, $P$ the law of a random variable $X$ and $\varphi$ is the characteristic function of $X$ $$ \int_{\mathbb R}f\ast\mathcal F^{-1}[\frac 1{\varphi}(-\bullet)](x)P(dx)=f(0) $$ for any uniformly bounded function $f$. […]

Fourier Analysis and its applications

This question already has an answer here: Fourier transform for dummies 14 answers

Complexity of FFT algorithms (Cooley-Tukey, Bluestein, Prime-factor)

I need to be able to explain the complexity of three Fast Fourier Transform algorithms: Cooley-Tukey’s, Bluestein’s and Prime-factor algorithm. Unfortunatelly, I’m a little lost in the process. Discrete Fourier Transform general formula \begin{align} x &= \{x_0, … , x_{N-1}\}\\ 0 &\leq n \leq N-1 \\ X_n &= \sum^{N-1}_{k=0}x_k \cdot e^{\frac{-i\cdot 2\pi \cdot k \cdot […]

Using Parseval's theorem to solve an integral

The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

Computing Fourier transform of power law

I’m trying to compute the Fourier transform of $$f(\mathbf{r}) = \frac{1}{r^\alpha}$$ where $\mathbf{r} \in \mathbb{R}^n$. For sufficiently large $\alpha$, the Fourier transform exists. One well-known example in physics is the case $\alpha = n-2$, which is the Coulomb potential; its Fourier transform is $1/k^2$. For a general $\alpha$, I can argue that $$\widetilde{f}(\mathbf{k}) \sim \frac{1}{k^{n-\alpha}}$$ […]

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert x\rvert\rightarrow\infty$$ and $u$ is bounded as $$y\rightarrow\infty$$ I’ve tried hard for this,but my answer is not matching.I’ve gone through by applying infinite fourier transformation as my text book suggests,but I’m not even getting […]